J. Phys. Chem. B 2001, 105, 11559-11565
11559
Considerations of the Isothermal-Isobaric Homogeneous Nucleation of a Vapor in the Presence of an Inert Carrier Gas† Wolfram Vogelsberger* Institute of Physical Chemistry, Chemistry and Earth Science Faculty, Friedrich-Schiller-UniVersity Jena, Lessingstrasse 10, D-07743 Jena, Germany ReceiVed: April 9, 2001; In Final Form: May 22, 2001
The nucleation process of a vapor in the presence of a carrier gas is investigated under isothermal-isobaric conditions. The constancy of the pressure in the system can in principle be maintained by a change of the volume of the system or the addition of the condensed amount of substance, respectively. The first possibility is used in several experiments and it is discussed in this contribution. The pressure of the carrier gas increases and the pressure of the condensing species decreases during the formation of nuclei, or droplets. The droplet formation causes a depletion of the mother phase. The initial amount of condensable species is distributed on droplets of identical size and having a determined concentration and a remaining vapor part in the frame of the model. A large number of such possibilities exist if the initial amount of condensable species is large. These states of the system are compared to the initial state of the system, the supersaturated vapor, by calculating the difference in the Gibbs free energy of nucleation. A free energy surface is obtained in the droplet sizedroplet concentration space. This surface shows maxima and minima that may be determined by Kelvin-like equations. The true bottom of the valley of the surface must be calculated by the determination of gradient curves. This bottom line connects the relatively stable states of the system. The Kelvin-like curves and the bottom line of the surface exhibit a nucleation phase and Ostwald ripening. The true stable state of the system will be obtained by all these curves. It is the equilibrium between the bulk liquid and the saturation vapor pressure. The considerations are applied in a simple kinetic model for droplet growth. By this way it is possible to determine the rate constant for the droplet growth without further assumptions. Good agreement is observed to experimental results.
1. Introduction Isothermal-isobaric conditions are mainly used to interpret homogeneous nucleation experiments of vapors. In this context the presence of an inert carrier gas is always allowed. The task of the carrier gas is to take the heat of condensation away and thus keep the temperature nearly constant in the nucleating system. The evaluation of the experiments is done under the assumption of constant pressure. Especially in the last years sophisticated experimental setups are developed which allow to keep the pressure constant after a nucleation period, see, for example, refs 1-5. This methodology incorporates a “nucleation pulse” to avoid coupling the nucleation event with the subsequent droplet growth/vapor depletion. At that point it is necessary to distinguish between the total pressure and the partial pressure of the condensing substance in the system. The appropriate thermodynamic potential for the nucleation under constant total pressure is the Gibbs free energy. The thermodynamic calculations of the Gibbs free energy are mostly done by the assumption of constant supersaturation in the system. The aim of this contribution is that there is a difference between the assumption of constant total pressure on one hand and constant supersaturation on the other hand. It †
Part of the special issue “Howard Reiss Festschrift”. * To whom correspondence should be addressed. Institut ikalische Chemie, Chemisch-Geowissenschaftliche Fakulta¨t, Schiller-Universita¨t Jena, Lessingstrasse 10, D-07743 Jena, Phone: +49 3641 948340. Fax: +49 3641 948302. E-mail: uni-jena.de.
fu¨r PhysFriedrichGermany. c9vowo@
will be shown that it is only possible to keep the total pressure constant by moving a piston in the appropriate way. The supersaturation cannot be kept constant by this procedure. The only way to maintain constant supersaturation would be the addition of the condensed amount of substance to the system. In this case the total pressure is also constant since the partial pressure of the inert gas does not change. A crucial point is the ratio of the pressure of the nucleating species to the pressure of the inert gas. If this ratio is large, i.e., high pressure of the nucleating species and the amount condensed in the nuclei is small, then it is justified to work with a constant supersaturation. On the other hand several papers have been published that indicate an influence of the carrier gas itself on the experimental results of nucleation experiments.6-9 In the present article the case of a small ratio, i.e., a large excess of inert carrier will be discussed. The considerations are made for the nucleation from the vapor phase and the formation of liquid or solid droplets. These considerations can easily be applied to the situation in a condensed phase if the appropriate expressions for the chemical potentials are introduced. 2. Thermodynamic Calculations In the present work a system is considered at constant temperature T and pressure pg. It is composed of two species: the condensable species of total amount of substance ng0(1) and the inert carrier gas ng(2). The upper index, 0, indicates the initial amount of substance in the system without condensation and the upper index, 1, is used if condensation has occurred. The
10.1021/jp011304q CCC: $20.00 © 2001 American Chemical Society Published on Web 07/18/2001
11560 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Vogelsberger
lower indices g and l indicate the vapor and the liquid phase, respectively. Indices are only used if they are necessary for distinction. The total amounts of substance of both species n0 is also constant
ng0(1) + ng(2) ) n0
(1)
If nucleation takes place under these conditions the internal energy of the system u and the volume V of the system must change.10 The initial state and a state after the nucleation has started is compared. The total pressure of the system may be written in the following manner
pg ) pg0(1) + pg0(2) ) pg1(1) + pg1(2) ) RGT 0 {ng (1) - nl(1) + ng(2)} (2) V1 RG is the gas constant. The partial pressures of both species have changed if condensation has occurred since the volume must change from V0 to V1. We can write
pg1(1) )
{ng0(1) - nl(1)}RGT{ng0(1) + ng(2)} V0{ng0(1) - nl(1) + ng(2)}
)
pg (1)
ng(2)RGT{ng0(1) + ng(2)} V0{ng0(1) - nl(1) + ng(2)}
{ }
µg (1) ) µg (1) + RGTln
ps(1)
n0
(3)
n0
µg1(1) ) µgs(1) + RGT ln
,
{ } pg1(1) ps(1)
{ } pg0(2) ps(2)
,
µg1(2) ) µgs(2) + RGT ln µl1(1) ) µls(1) + Vl(1){pl1(1) - ps(1)}
)
(
0
σol ng (1)RGT
- ln y -
ng0(1)
{ } pg1(2) ps(2)
(4)
Vl(1) is the molar volume of the condensed phase. The upper index s is used for the standard state. The appropriate thermo-
+
nl(1)
1-
n0 ng (1)nl(1) 0
ng (2) ln 1 - 0 0 ng0(1) n ng (1)
)
+
}
[pl1(1) - ps(1)]
(6)
y ) ng0(1)/ngs(1) is the initial supersaturation and ngs(1) is the amount of substance generating the saturation vapor pressure ps(1) in the system. The last term in eq 6 is small compared to the other terms and it is therefore neglected as it is frequently done. The calculations are made not only for one single cluster in the system but for a number of clusters, Z, having a radius, R. The clusters are assumed to be droplets. The following dimensionless quantities are introduced:
∆ Kg 0
ng (1)RGT
) g(r,z),
z)
nl(1) 0
)
ng (1)
4πNAR3
Z ) zr3 3Vl(1) Ng0(1)
( )
4πNA Z , r)R 0 3Vl(1) Ng (1) h)
µg0(2) ) µgs(2) + RGT ln
ng0(1)
ln y
nl(1)
nl(1)
1 nl(1)
{ }
nl(1)
ng0(1)RGT
Upon considering the expressions in eq 3, it is evident that the partial pressure of the nucleating species, 1, decreases during the condensation process and the partial pressure of the carrier, 2, increases. If ng(2) . ng0(1), it would be possible to work with a constant chemical potential of the carrier. This, however, is improper because it leads to some thermodynamic inconsistencies. The chemical potentials can be written as
pg0(1)
{ [ ( )] (
∆Kg ) ng0(1)RGT 1 -
1-
nl(1)Vl(1)
{ } 1-
s
σ is the surface tension of the condensed phase. Introduction of the expressions for the chemical potentials into eq 5 with regard to eq 3 leads to the following equation for ∆Kg:
ng0(1)
) pg(2)
0
σol - ng0(1)µg0(1) - ng0(2)µg0(2) (5)
nl(1)
1-
pg1(2) )
∆Kg ) ng1(1)µg1(1) + ng1(2)µg1(2) + nl(1)µl1(1) +
0
1-
0
dynamic potential for an isotherm-isochoric condensation process is the Gibbs free energy of condensation ∆Kg. It is a first-order homogeneous function of the extensive variables n(i), and the surface area of the condensed phase ol.11 It may be calculated by the difference in Gibbs free energy between the final state, system with clusters, and the supersaturated initial system composed only of vapor. We can write
1/3
ng0(1) 4πσ 2 R0 , a ) 0 kBT n
)
R R0 (7)
NA is the Avogadro constant, and Ng1(1) is the total number of molecules of species 1 in the system. The number of clusters Z is related to Ng1(1), and the radius of the clusters is related to the radius of a molecule R0, if the molecule is taken to be a sphere. We can write
g(r,z) ) (1 - zr3) ln{y(1 - zr3)} 1 (1 - azr3) ln{1 - azr3} + hzr2 - ln y (8) a If no depletion of the nucleating species would occur, the supersaturation would remain constant. The term zr3 in the logarithms of eq 8 vanishes and ln(y) cancels with one in the first bracket. In this case the relation for the Gibbs free energy mainly used in nucleation theory would remain.
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11561
Figure 1. Gibbs free energy of nucleation g(r,z) as a function of cluster size r and cluster concentration z.
The Gibbs free energy of condensation according to eq 8 is shown in Figure 1. The calculations are made for T ) 273 K (σ ) 75,6 mN m-1, Vl(1) ) 18.018 cm3 mol-1)12 and a supersaturation y ) 2. In this case the partial vapor pressure of water is ps(1) ) 610.38 Pa.12 If the nucleation of water in air is considered under atmospheric conditions this pressure is small compared to the carrier (air). The supersaturation is chosen in such way that the mean features of the Gibbs free energy surface are in the same order of magnitude and therefore visible in the representation Figure 1. The Gibbs free energy of condensation depends on cluster size and cluster concentration, g(r,z), and forms a free energy surface. The main difference to the common treatment in nucleation theory is the depletion of the water vapor during the cluster formation. Therefore, the actual supersaturation changes continuously with proceeding cluster formation and growth. The surface shows a crest and a valley. These extrema of g(r,z) are found in the common manner by setting the appropriate derivatives equal to zero
vapor supersaturation in the system. The denominator in the logarithms does not occur in the classical treatment for only one component under isothermal-isobaric conditions.10 If there is a large excess of carrier (n0 . ng0(1)), the denominator in eqs 9 becomes one. Equation 9a is evidently the classical Kelvin equation if no depletion of the vapor takes place, zr3 ) 0. The features of the surface of the Gibbs free energy in dependence on droplet size and droplet concentration can be explained more impressively if a contour plot of this surface is calculated, Figure 2. At the first it is evident that there is a limit of the range of definition of eq 8. This limit is given by the graph zr3 ) 1. In this case the whole amount of the condensable species 1 ng0(1) is contained in droplets of the appropriate size and concentration. This of course is a thermodynamically improbable state. The Gibbs free energy, g(zr3)1) is
∂g(r,z) y(1 - zr*3) 2h - ln )0) ∂r z 3r* 1 - azr*3
(9a)
∂g(r,z) y(1 - zr×3) h ) 0 ) × - ln ∂z r r 1 - azr×3
(9b)
The actual supersaturation yact in the system can change from the initial value y to zero if all molecules of the condensable species are incorporated in the clusters. Numerical examples are given for the situation considered (y ) 2, T ) 273 K). If the droplet radius is 20 and the concentration is 10-5 as starting configuration then the constant a is a ) 0.0122 and yact ) 1.8418 is calculated. Increase of the droplet radius to 30 results in yact ) 1.4648, r ) 40 gives yact ) 0.7257 (already under saturated).
( ) ( )
{ {
} }
The argument of the logarithms in eqs 9 is the actual water
h 1 g(zr3)1) ) - (1 - a) ln(1 - a) - ln y r a
(10)
11562 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Vogelsberger
Figure 2. Contour plot of the Gibbs free energy of nucleation g(r,z), small numbers near the contour lines, as a function of cluster size r, and cluster concentration z. For detailed explanation, see text.
Increasing the droplet concentration in the start configuration to 5 × 10-5 leads to yact ) 1.2059. The influence of the constant a is generally small. If it is neglected in the starting configuration yact ) 1.84 is calculated. The green line labeled A represents eq 9a. As stated before it is the analogue to the Kelvin equation if depletion of the mother phase is allowed. It starts at the radius r*(zf0) ) 8.99. This graph shows a maximum. Radii smaller than the maximum belong to maxima of g(r,z), i.e., critical states. Radii larger than the maximum belong to minima of g(r,z), i.e., relatively stable states of the system. Thus, it may be concluded that in the z-region below the maximum of curve A two radii exist for one given cluster concentration (z ) constant). One radius belongs to a maximum and the other to a minimum of g(r,z). If the cluster concentration exceeds the maximum of curve A, no extrema exist. The broken green line labeled B represents eq 9b. It starts at the radius r×(zf0) ) 13.48, and it also shows a maximum. Below this maximum two radii r× can be determined at z ) constant. They belong to two minima of g(r,z). If the cluster
radius tends to infinity, rf∞, both graphs A and B, respectively, approach again zf0. This is the equilibrium of the bulk condensed phase at saturation vapor pressure. The maximum value of the condensed amount of substance under equilibrium conditions is therefore zr3 ) {ng0(1) - ngs(1)}/ng0(1) ) (1-1/ y). The corresponding value of g(r,z) is the Gibbs free energy of the bulk liquid (r ) ∞), gb, is
lim
zr3f(1 - 1/y)
g(r,z) ) gb ) - ln y -
{
)} {
1 1 1 1 - a 1 - ln 1 - a 1 a y y
(
(
)} (11)
This is the only thermodynamically stable state of the system. The driving force for variations in thermodynamic systems is a change in the potential function, i.e., the determination of the gradient of g(r,z) in the system considered.
∇g(r,z) )
(
) (
)
∂g(r,z) ∂g(r,z) e + e ∂r z r ∂z r z
(12)
Homogeneous Nucleation of a Vapor
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11563
er and ez are the unit vectors in the r- and z-directions, respectively. The gradient curves in the contour plot are given by the following relation:13
( (
) )
∂g(r,z) dz ∂z ) dr ∂g(r,z) ∂r
r
(13)
z
Equation 13 cannot be solved analytically as it has been the case if only the condensing species is present in the system.10 The gradient curves have to be determined by numerical integration or by an appropriate mathematical procedure, which calculates the gradient curve from point to point at the surface. The later possibility is applied in our case. The calculated curves are the red lines labeled C and D and the blue curves labeled by the numbers 1 to 7. The red curve C starts at the radius r×(zf0) ) 13.48. First it develops quickly in the direction of decreasing Gibbs free energy and of larger droplet concentration and droplet radius. This period may be called a nucleation phase. Then curve C passes a maximum and approaches the equilibrium of bulk liquid at saturation vapor pressure by increase of the droplet size and decrease of their concentration. This period may be called Ostwald ripening. The start point of curve D may also be at the radius r×(zf0) ) 13.48. From here the curve develops in the direction of the increasing Gibbs free energy and of smaller radii and increasing droplet concentration. After passing a minimal radius curve D went again in the direction of larger radii, whereas the concentration of droplets increases further. There is a considerable region where curves A and D are close to each other. Curve D ends if the whole amount of substance is incorporated in droplets (zr3 ) 1), eq 10, the limit of the range of definition of g(r,z). Curve D separates the gradient curves in two groups. Graphs 1 to 3 end either on the z- or r-axes, respectively. That means that either the radius or the concentration of the droplets becomes zero. Droplets on the left-hand side of curve D are therefore instable and tend to decompose. Looking carefully at graphs 2 and 3 shows that they seem to have a special property near the abscissa. Graph 2 turns in the direction to the origin of the coordinate system and graph 3 turns to r×(zf0) ) 13.48. This behavior is not being observed in the isobaric condensation of a single component where eq 13 can be solved analytically.10 This could mean that the classical Kelvin radius, r*(zf0) ) 8.99, is the border between decomposition and growth of one droplet. This conclusion is obvious if only one droplet is considered, but this is the constraint of a fixed droplet concentration. Therefore, the behavior of the gradient curves in this size-concentration region has a special meaning. The gradient curves on the right-hand side of curve D, graphs 4 to 7 all tend to quickly approach curve C. At larger radii they coincide with curve C. Therefore, curve C is the bottom line of the g(r,z) surface. States of the system placed on curve C are relatively stable. The greater the stability, the smaller the gradient of g(r,z). This is the case for larger droplets. It is therefore to understand that a fog or a precipitate is relatively stable without an additional stabilization. It is also worth mentioning that curves A and B approach curve C for larger droplets. The analogue of the Kelvin equation, curve A, describes critical and relatively stable states of the system not generally in an appropriate manner. But there are regions where it works well. The relation between droplet size and droplet concentration given by curves A, B, and C is qualitatively established experimentally.14
3. Kinetic Calculations Now it is tried to give a kinetic description of the formation of small clusters or droplets using the described thermodynamic considerations. As stated before the nucleation process may start with one single droplet. Larger concentrations would reduce in the size region from r ) 0 to r ) r*(zf0) ) 8.99. The droplet has to overcome the critical size r*(zf0) ) 8.99 and then to grow up to r×(zf0) ) 13.48. The classical nucleation model could describe this first period. Starting at r×(zf0) ) 13.48 the system should follow curve C which is the bottom line of the g(r,z) surface. Curve C can be determined only numerically in the present case. Probably a simplification is possible and to replace the true bottom by curve B. In this case an analytic expression for the dependence of droplet concentration on droplet size can be obtained by a rearrangement of eq 9b. This simplification is applied and it works well for larger radii. The development of the system in time may be described by the following reaction equation kf
} Ci+1 Ci + M {\ k
(14)
d
i indicates the number of monomer units in the droplet and kf and kd are the rate constants of monomer addition or abstraction, respectively. This reaction scheme is mostly used in kinetic description of nucleation processes, see, for example, refs 15 and 16. From this reaction scheme the following rate equation is evident:
{
}
4πNS 2 [M] d[M] ) kf[C][M]s 1 , [C] ) ZR (15) s dt VNA [M] Brackets indicate concentrations. The rate constant kd is replaced in eq 15 by the equilibrium condition d[M]/dt ) 0. [M]s is the saturation concentration of the vapor. It is assumed that the droplet concentrations of two neighboring states at the g(r,z) surface are equal, [Ci] ) [Ci+1] ) [C]. Only the surface atoms of the droplets can act as condensation centers in the sense of eq 14. The concentration [C] may therefore be replaced by the second expression in eq 15. NS is the number of surface atoms per unit of surface area. Equation 15 may be transformed by the introduction of the amount of liquid of the condensable species and by use of eqs 7 and 9b to give
dnl(1) ) kf dt σ
(
NSkBT V0
ng0(1)
- Vl(1)
{[ } [
y 1-
)
nl(1) ln
nl(1)
ng0(1)
] ]
nl(1) 1-a 0 ng (1)
{
×
}
nl(1) 1 1- - 0 y n (1) g
(16)
The actual volume of the gas phase changes insignificantly by the formation of the liquid. This is accounted for in eq 16. The change of the radius of the droplets in time can be obtained from eq 16
( )
dR dnl(1) dnl(1) ) dt dt dR
-1
(17)
The derivative dnl(1)/dR may be calculated by a rearrangement of eq 9b (zr×3 ) nl×(1)/ng0(1)). Combination of eqs 16 and 17
11564 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Vogelsberger
Figure 3. Comparison of experimental determined rate of growth of water droplet radius R as a function of time t. Squares indicate experimental results. The line is obtained by numerical integration of eq 20.
results therefore in
dR ) kf dt σ
[
NSkBT V0
ng0(1)
]
R{ye-hR0/R - 1} ×
- Vl(1) (1 - a)y
{(1 + ay - a)e
hR0/R
}
- 1 (18)
A very simple formula can be obtained if the volume of the droplets is neglected compared to the total volume (V0 . Vl), and the amount of the condensing species ng0(1) is small compared to the carrier (a , 1). It is further assumed that the droplet concentration is constant. This assumption is justified if the droplets are relatively large and the range of droplet size investigated is small. In this case the derivative dnl(1)/dR has to be taken at constant droplet concentration
( )
dR dnl(1) dnl(1) ) dt dt dR
-1
z
(19)
The combination of eqs 7, 15, and 19 leads to
NSVl(1)ns(1) hR0/R dR ) kf {e - 1} ) kdg{ehR0/R - 1} (20) 0 dt z VN
( )
A
Careful experimental results on the growth of water droplets are reported.5 Equation 20 can be solved numerically and the result is compared to these experiments. The constant hR0 is calculated for the experimental conditions, T ) 266.5 K, σ ) 77.13 × 10-3 Nm-1, and Vl (1) ) 18.037 cm3 mol-1,17 and the integration is carried out beginning from the first experimental point which is taken to be without error. The comparison is demonstrated in Figure 3. A droplet growth rate constant kdg ) 87.33 µm/ms has been determined and the correlation coefficient is 0.999671. This procedure is the common way to determine kinetic rate constants. In view of this, a surprisingly good agreement is found between experiment and theory. 4. Conclusions The nucleation of a vapor in the presence of a carrier is investigated under isothermal-isobaric conditions. It is possible to keep the pressure constant by two ways. The condensed amount of substance must be replaced in the system. In this case the supersaturation remains also constant. The other possibility is to move a piston in the appropriate manner to
decrease the volume and thus keep the pressure unchanged. This possibility is used in several experiments. In this case it is evident that the pressure of the nucleating species decreases and the pressure of the carrier increases. Therefore, the supersaturation changes continuously if the formation of the new phase proceeds. This process is investigated thermodynamically. The system considered consists of a fixed amount of condensable species. The amount of substance condensed is distributed on droplets of uniform size. The concentration of the droplets is therefore determined. If the amount of substance is large, there exist a large number of possibilities to distribute this amount of substance on droplets of different size. All the states of the system obtained by this manner are compared to the initial supersaturated system without liquid. For this aim the difference in the Gibbs free energy of nucleation is calculated. The Gibbs free energy of nucleation forms a surface in the size concentration space. This surface shows extrema. Derivation of the Gibbs free energy of nucleation with respect to the droplet radius at constant droplet concentration leads to a relation between droplet size and actual supersaturation. This relation transforms in the well-known Kelvin equation with continuous change of the supersaturation if the amount of carrier is much larger than the amount of the condensable species. Two radii exist for each droplet concentration, which are solutions of the relation between droplet size and actual supersaturation, eq 9a. One solution belongs to a maximum of the Gibbs free energy of nucleation and the other one belongs to a relative minimum. An analogous equation to this equation can be derived if the derivation of the Gibbs free energy is carried out at constant droplet radius, eq 9b. Both equations differ by the factor 3/2 in the nonlogarithmic term (eqs 9). These results may be compared to the isothermal isobaric nucleation without the presence of a carrier gas.10 In this latter case the pressure of the nucleating species itself is constant. No depletion can occur. Only one critical droplet radius exists independent of the number of droplets considered. It is determined by the Kelvin equation. No extremum exists if the Gibbs free energy of nucleation is derived with respect to the number of droplets at constant droplet radius. In this case the radius r× determines the position where the Gibbs free energy of the system droplets-vapor is zero, i.e., it has the same value as that in the initial supersaturated vapor. It is possible to calculate gradient curves at the Gibbs free energy surface, i.e., the determination of the steepest decent of the potential function. The gradient curves may be divided into two groups. If the chosen state of the system is characterized by a droplet size and droplet concentration on the left-hand side of curve D in Figure 2, the droplets tend to decompose. If the position of the system is on the right-hand side of curve D, the gradient curves approach the curve labeled C. Curve C is therefore the bottom of the Gibbs free energy surface. It determines relatively stable states of the system. The first part of curve C up to the maximum is a nucleation phase since growth of the droplets is accompanied by increasing droplet concentration. After the maximum the droplet growth is accompanied by a decrease of their concentration. This is the phase of Ostwald ripening. If the system is chosen to be large enough, the bottom of the Gibbs free energy surface ends at the equilibrium of bulk liquid at saturation vapor pressure. If a small system is investigated, the end point of the condensation process is achieved if one droplet exists at its equilibrium vapor pressure. Gravitational effects are excluded. This is the point where phenomenological thermodynamic considerations and molecular theory18,19 meet each other. In this context it is worth
Homogeneous Nucleation of a Vapor mentioning that the influence of the carrier gas in theoretical investigations of the nucleation theorem is also considered.20,21 The relation between droplet size and actual supersaturation, curve A in Figure 2, is the analogue to the common Kelvin equation if the ratio of carrier and condensing species is large in the system considered. This equation or the Kelvin equation, respectively, is not generally suitable to determine critical and relatively stable states of the system. But there are regions where they work well. A simple kinetic model is presented for the droplet formation in the system under investigation. It bases on the thermodynamic considerations. By this model the rate constant of droplet growth can be determined. Comparison to experimental results of droplet growth shows surprisingly good agreement. Acknowledgment. The author thanks Dr. Mario Lo¨bbus for his support in preparing the graphics. The graphics are made by the help of Mathematica (Wolfram Research). References and Notes (1) Schmitt, J. L. ReV. Sci. Instrum. 1981, 52 (11), 1749. (2) Wagner, P. E.; Strey, R. J. Phys. Chem. 1981, 85, 2694. (3) Miller, R. C.; Anderson, R. J.; Kassner, J. L., Jr. Hagen, D. E. J. Chem. Phys. 1983, 78 (6), 3204.
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11565 (4) Viisanen, Y.; Strey, R.; Reiss, H. J. Chem. Phys. 1993, 99 (6), 4680. (5) Rodemann, T.; Peters, F. J. Chem. Phys. 1996, 105 (12), 5168. (6) Heist, R. H.; Janjua, M.; Ahmed, J. J. Phys. Chem. 1994, 98, 4443. (7) Heist, R. H.; Ahmed, J.; Janjua, M. J. Phys. Chem. 1995, 99, 375. (8) Wyslouzil, B. E.; Wilemski, G.; Beals, M. G.; Frish, M. B. Phys. Fluids 1994, 6, 2845. (9) Kane, D.; Fisenko, S. P.; Rusyniak, M.; El-Shall, M. S. J. Chem. Phys. 1999, 111 (18), 8496. (10) Vogelsberger, W. Z. Phys. Chem. 2001, 215. In press. (11) Mu¨nster, A. Chemische Thermodynamik; Akademie-Verlag: Berlin, 1969, p 84. (12) Handbook of Chemistry and Physics, 63rd ed.; CRC Press: Boca Raton, Florida, 1982-1983. (13) Vogelsberger, W.; Beck, M.; Fritsche, M.; Ma¨urer, F. Z. Phys. Chem. 1992, 175, 201. (14) Stein, G. D.; Wegener, P. P. J. Chem. Phys. 1967, 46 (9), 3685. (15) Dillmann, A.; Meier, G. E. A. J. Chem. Phys. 1991, 94 (5), 3872. (16) Delale, C. F.; Meier, G. E. A. J. Chem. Phys. 1993, 98 (12), 9850. (17) Rodemann, T. Ph.D. Thesis, Universita¨t GH Essen, Essen, Germany, 1996. (18) Vogelsberger, W. Chem. Phys. Lett. 1980, 74 (1), 143. (19) Senger, B.; Schaaf, P.; Corti, D. S.; Bowles, R. K.; Voegel, J.-C.; Reiss, H. J. Chem. Phys. 1999, 110 (13), 6421. (20) Oxtoby, D. W.; Laaksonen, J. Chem. Phys. 1995, 102 (17), 6846. (21) Bowles, R. K.; McGraw, R.; Schaaf, P.; Senger, B.; Voegel, J.-C.; Reiss, H. J. Chem. Phys. 2000, 113 (11), 4524.
